Answer to Question #248729 in Physics for James Andrew

Question

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement using the component method.

Answer 1

Let's first find $x$- and $y$-components of the resultant displacement:

$d_x=3.25\ km\cdot cos90^{\circ}+2.20\ km\cdot cos180^{\circ}+1.5\ km\cdot cos270^{\circ}=-2.2\ km,$

$d_y=3.25\ km\cdot sin90^{\circ}+2.20\ km\cdot sin180^{\circ}+1.5\ km\cdot sin270^{\circ}=1.75\ km.$

We can find the magnitude of the resultant displacement from the Pythagorean theorem:

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(-2.2\ km)^2+(1.75\ km)^2}=2.81\ km.$$

We can find the direction of the resultant displacement from the geometry:

$$\theta=tan^{-1}({\dfrac{d_y}{d_x}}),$$ $$\theta=tan^{-1}({\dfrac{1.75\ km}{-2.2\ km}})=38.5^{\circ}\ N\ of\ W.$$